##
**Regular and chaotic oscillations.**
*(English)*
Zbl 0980.70001

Foundations of Engineering Mechanics. Berlin: Springer. xii, 395 p. (2001).

The modern theory of nonlinear oscillations has been largely developed in the former UDSSR during the first half of the last century. Such prominent scientists as Andronov, Mandelshtam, Krylov, Bogolyubov and many others have made significant contributions. These works have been excellently presented and documented in the classic book by A. A. Andronov, A. A. Witt and S. E. Chaikin [Theory of oscillators. International Series of Monographs in Physics. 4. Oxford etc.: Pergamon Press. xxxii (1966; Zbl 0188.56304)]. In the second half of the last century, due to the progress achieved by dynamical system theory, a more mathematical foundation of these earlier works was given. For example, new concepts developed in bifurcation theory, which to some extend were intuitively anticipated by Andronov and his group, allowed a more rigorous classification of oscillation phenomena. A typical example is the flutter-type of loss of stability of equilibrium which now usually is called Hopf bifurcation. More correctly, it should be called Poincaré-Andronov-Hopf bifurcation. Other “new” concepts, like chaos (also already known to Poincaré) opened new domains of research. Hence the field of nonlinear oscillations preserves up today a very dynamic character.

The present book reflects this situation very well and gives an up todate overview of recent developements, especially focussing on work done in the countries of the former Soviet Union. In this respect it is a gold mine for readers who have already some familiarity with the theory of nonlinear oscillations, because 375 references are listed.

The well-known author who is a world-wide recognized expert in the field has herself made strong contributions to the field. In this book she restricts to dissipative oscillation phenomena, always stressing the physical properties of the considered phenomenon. Precise references allow the interested reader to find more details. Besides classical fields like natural, self-excited and parameter-excited oscillations, also modern researchfields like synchronization, chaotic and stochastic oscillations are treated in detail. The author discusses various applications ranging from biological to electrical and mechanical systems, and also addresses infinite-dimensional problems.

However, the interested reader must be aware that this book is not a textbook for beginners, even if it could be of some interest to newcomers to the field, because of the great variety of phenomena presented. For those readers who want to get an overview on modern developements of oscillation theory and their applications in the sciences, the book will be certainly an excellent choice.

The present book reflects this situation very well and gives an up todate overview of recent developements, especially focussing on work done in the countries of the former Soviet Union. In this respect it is a gold mine for readers who have already some familiarity with the theory of nonlinear oscillations, because 375 references are listed.

The well-known author who is a world-wide recognized expert in the field has herself made strong contributions to the field. In this book she restricts to dissipative oscillation phenomena, always stressing the physical properties of the considered phenomenon. Precise references allow the interested reader to find more details. Besides classical fields like natural, self-excited and parameter-excited oscillations, also modern researchfields like synchronization, chaotic and stochastic oscillations are treated in detail. The author discusses various applications ranging from biological to electrical and mechanical systems, and also addresses infinite-dimensional problems.

However, the interested reader must be aware that this book is not a textbook for beginners, even if it could be of some interest to newcomers to the field, because of the great variety of phenomena presented. For those readers who want to get an overview on modern developements of oscillation theory and their applications in the sciences, the book will be certainly an excellent choice.

Reviewer: H.Troger (Wien)

### MSC:

70-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems |

70Kxx | Nonlinear dynamics in mechanics |

70L05 | Random vibrations in mechanics of particles and systems |

74H45 | Vibrations in dynamical problems in solid mechanics |

74H50 | Random vibrations in dynamical problems in solid mechanics |

74H65 | Chaotic behavior of solutions to dynamical problems in solid mechanics |